Question

If $ {{S}_{n}}=\frac{1}{6.11}+\frac{1}{11.16}+\frac{1}{16.21}+.... $ to $ n $ terms, then $ 6{{S}_{n}} $ equals

• A. $ \frac{5n-4}{5n+6} $
• B. $ \frac{n}{(5n+6)} $
• C. $ \frac{2n-1}{5n+6} $
• D. $ \frac{1}{(5n+6)} $

Answer 1

Answer: (B)

Given that, $ {{S}_{n}}=\frac{1}{6.11}+\frac{1}{11.16}+\frac{1}{16.21}+....+n $ terms $ =\frac{1}{5}\left( \frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+....\frac{1}{5n+1}-\frac{1}{5n+6} \right) $ $ =\frac{1}{5}\left( \frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+....+\frac{1}{5n+1}-\frac{1}{5n+6} \right) $ $ =\frac{1}{5}\left( \frac{1}{6}-\frac{1}{5n+6} \right) $ $ =\frac{1}{5}\left( \frac{5n+6-6}{6(5n+6)} \right) $ $ =\frac{n}{6(5n+6)} $ $ \Rightarrow $ $ 6{{S}_{n}}=\frac{n}{5n+6} $